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Questions tagged [brachistochrone-problem]

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the problem of finding the path between two points such that the transit time under specified conditions is minimized.

55 questions
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5 votes
2 answers
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Short version: Before solving the problem (in particular, before identifying the cycloid as the only possible minimizing curve), is there any physical intuition why we should expect there to be a ...
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3 answers
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I am studying a physics problem where an object moves from one point to another under frictionless conditions. The trajectory of the object is shaped like a downward parabola, but the exact curvature ...
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Is there a solution to the brachistochrone problem where the solution is non-differentiable everywhere (angular point)? The Euler-Lagrange method fails if the first or second derivative of the ...
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I am looking for an ansatz of the following problem: Given a mass $m$ moving in a constant gravitational field along curves $C$ connecting two fixed points I want to find the curve $C_0$ that ...
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5 votes
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The Brachistochrone problem is usually presented with the having a ball dropped into the slide with initially zero velocity and at position $(x, y)=(0, 0)$. I would like to know the more general ...
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7 votes
1 answer
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I wonder how I can solve the Brachistochrone problem for 3 points? The matter starts from point A that is the highest point and it must pass from B and must finish with point C. (No any friction in ...
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2 votes
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Brachistochrone problem The time to travel from point $p_1$ to $p_2$ is given by this integral $$t_{12}=\int_{p_1}^{p_2}\frac{ds}{v}.$$ With $ds=\sqrt{dx^2+dy^2}=\sqrt{1+y'^2}\,dx$ and $v=\sqrt{2g\,y}$...
Eli's user avatar
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1 vote
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Given 3 ways of going downhill, like in this image: Would a ball behave like that in real life? Intuitively, it makes no sense. The shortest path here is not the fastest. Any hints to the math ...
Alwaz's user avatar
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The original Brachistochrone problem is without friction and drag. The Brachistochrone problem can also be solved analytically with friction. But what would the optimal path be if there was a drag ...
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2 votes
1 answer
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I'm trying to understand this solution to the brachistochrone problem inside a uniform sphere. Going from equation (19) to (20) and (21), the integrand of the functional we're trying to minimize is of ...
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I am reading a differential equation book that discusses the Brachistochrone problem. The book discusses Bernoulli's solution that uses Snell's law. The book says that a ray would follow the fastest ...
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Recalling the statement of the problem : Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the ...
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When I came across the Brachistochrone problem, my teacher said we could relate it to Fermat's principle of least time. So, we could make many glass slabs of high $\mathrm dx$, and every slab has a ...
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Given two points in space, the 2D Brachistochrone problem could be solved to give solution of a cycloid. I am wondering how could one prove that in arbitrary dimensions ($d\geq 3$) with a 1D uniform ...
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There are two blocks, each starting at the top of an incline. The particular inclines are depicted in the image below. The height through which the blocks fall is the same, the table lengths are the ...
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